Tamaiko Chappell

FINAL ASSIGNMENT

Part A: Consider any triangle ABC.
Select a point P inside the triangle and draw lines AP, BP, and
CP extended to their intersections with the opposite sides in
points D, E, and F respectively.

Consider the following triangle:

*Explore (AF)(BD)(EC)
and (FB)(DC)(EA) for various triangles and various locations of
P.

We'll examine a few different triangles
with different side measures and change the location of P for
each. For each of these we will compare the products of
(AF)(BD)(EC) and (FB)(DC)(EA) and see of there is any relationship:

EXAMPLE 1:

EXAMPLE 2:

EXAMPLE 3:

From these examples,
we can see that no matter the triangle nor the position of P inside
the triangle, (AF)(BD)(EC) and (FB)(DC)(EA) are always equal.

Part B: Conjecture? Prove it!

Because (AF)(BD)(EC) and
(FB)(DC)(EA) are always equal, I make the conjecture that the
ratio of these to products is equal to 1.

In order to prove this conjecture, I will
need to use similar triangles.

Proof: Make two lines that are parallel
to segment BD through points A and C.

Because the lines are parallel, I can use
the Alternate Interior Angles Theorem and the Vertical Angles
Theorem to get similar triangles.

I know that triangles EPC and AGC are similar
and triangles AEP and ACH are similar.